Bilbao Crystallographic Server Transformation matrix

The transformation matrix

• The matrix P

This matrix is often referred to as the linear part of the coordinate transformation and it describes a change of direction and/or length of the basis vectors.

The matrix P is a (3x3) matrix



P =		P11 P12 P13 P21 P22 P23 P31 P32 P33


and relates the new basis (a', b', c') to the old basis (a, b, c) according to



		P11	P12	P13
(a', b', c') = (a, b, c)P = (a, b, c)		P21	P22	P23		=	(aP11+bP21+cP31, aP12+bP22+cP32, aP13+bP23+cP33)
		P31	P32	P33

• The origin shift p

The origin shift is described by the shift vector p=p₁a + p₂b + p₃c. The coordinates of the new origin O^' with respect to the old coordinate sysem (a,b,c) are given by the (3x1) column



		p1
p=		p2
		p3




• Concise form

The data on the matrix-column pair (P, p) are often written in the following concise form:

P₁₁a+P₂₁b+P₃₁c, P₁₂a+P₂₂b+P₃₂c, P₁₃a+P₂₃b+P₃₃c; p₁,p₂,p₃

P = 1 1 0 -1 1 0 0 0 2

0 and p= 0 1/2

Euler angles

R =

Cosθ -Sinθ 0 Sinθ Cosθ 0 0 0 1

1 0 0 0 Cosφ -Sinφ 0 Sinφ Cosφ

CosΨ 0 SinΨ 0 1 0 -SinΨ 0 CosΨ